%I #13 Sep 12 2021 16:25:27
%S 0,2,3,4,10,12,21,24,36,50,66,84,117,140,180,224,289,342,437,520,630,
%T 770,920,1104,1300,1560,1809,2156,2523,2940,3441,3968,4620,5338,6125,
%U 7092,8103,9272,10608,12080,13776,15624,17759,20064,22680,25622,28858,32496,36456,40950,45849,51324,57399,64044,71390
%N Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).
%C Sum of all parts of all partitions of n into prime parts.
%C Convolution of the sequences A000607 and A008472.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimePartition.html">Prime Partition</a>
%H <a href="/index/Par#partN">Index entries for related partition-counting sequences</a>
%F G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 - x^prime(k)) * Product_{k>=1} 1/(1 - x^prime(k)).
%F G.f.: x*f'(x), where f(x) = Product_{k>=1} 1/(1 - x^prime(k)).
%F a(n) = n*A000607(n).
%F a(n) ~ n*exp(2*Pi*sqrt(n/log(n))/sqrt(3)).
%e a(6) = 12 because we have [3, 3], [2, 2, 2] and 2*6 = 12.
%t nmax = 55; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 - x^Prime[k]), {k, 1, nmax}] Product[1/(1 - x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]]
%t nmax = 55; Rest[CoefficientList[Series[x D[Product[1/(1 - x^Prime[k]), {k, 1, nmax}], x], {x, 0, nmax}], x]]
%t Table[Total@Flatten[IntegerPartitions[n,All,Prime@Range@PrimePi@n]],{n,52}] (* _Giorgos Kalogeropoulos_, Sep 12 2021 *)
%Y Cf. A000607, A008472, A066186, A084993.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Apr 10 2017